Why can any two phases be connected together?

The way I understood it and that made sense when connecting two phases together was that one of the phases was in the exact opposite range(polarity). So one phase would allow current to enter the circuit and then the other phase would allow the current back out of the circuit, back to the source. So it appeared to me that only certain combinations of 2 phases would work.

Well, then I read that ANY two phases could be connected together which totally blows that theory up.

How come a phase that's only 180' will add up voltage the same as the phase that's a whole 360' in the negative range?

I guess any phase to neutral voltage times the square root of 3 will equal the two phases. But why is this true for any phase combination?

How does two phases complete a circuit for current to travel?
I thought that the 180' would bring in because current is traveling in one direction(positive scale) and then the -180' would take it back as current would then be traveling in the opposite direction(negative scale)?

See, that's not how I thought it was suppose to work. I thought it was like this.
One phase is 100V at 120 and another phase is -100V at 120.
That makes sense; I see one side as being the positive and the other as being the negative. So initially I thought, yes we can use two phases but only when those two phases are fully inverse of each other. This appears to me as being the way that current can be carried back and forth throughout the cycles. One phase is coming in positive while another phase is going out negative.

But instead I am now finding out that we can also have 0V at 120(one of the phase wires) and 100V at 120(the second phase wire) or -100V at 120 and 0V at 120, which doesn't explain how current travels back to the source.

EDIT, wups; I'm not trying to use 120. I'm trying to use the full 360 degree range.
The top wave will be at 180'.
The bottom wave is at -180'.

See, that's not how I thought it was suppose to work. I thought it was like this.
One phase is 100V at 120 and another phase is -100V at 120.
That makes sense; I see one side as being the positive and the other as being the negative. So initially I thought, yes we can use two phases but only when those two phases are fully inverse of each other. This appears to me as being the way that current can be carried back and forth throughout the cycles. One phase is coming in positive while another phase is going out negative.

But instead I am now finding out that we can also have 0V at 120(one of the phase wires) and 100V at 120(the second phase wire) or -100V at 120 and 0V at 120, which doesn't explain how current travels back to the source.

EDIT, wups; I'm not trying to use 120. I'm trying to use the full 360 degree range.
The top wave will be at 180'.
The bottom wave is at -180'.

For simpicity we don't refer to the sinusoidal aspect of the system because we are assuming a contstant frequency. The voltage on each phase is described by the following functions:

P1 = V sin(ft + 0)
P2 = V sin(ft + 120)
P3 = V sin(ft - 120)

Where V is the peak Voltage, f is the frequency and t is time. If you plot these functions against time you get the diagram you showed earlier.

In the picture, we have a wave that is at peak for one phase. It's at 180. The three waves are displaced 120 degrees that's why they are at different heights at any given instant.
The one phase is at it's positive range meaning that voltage is positive.
There is another phase that is at a negative range so voltage is negative.

What I thought this showed was that the one phase that's positive is kind of like our hot and the phase that's negative is kind of like our return for current since it's going in the opposite direction. This was how I believe current was forced to travel through a circuit.

But when a phase that is near zero is being introduced;
that one phase can be at it's positive and another can be near zero in height - put together doesn't explain how current can follow a complete path back to the source. Voltage is pushing in, but no voltage is pulling out of the other phase; for example two phases connected to a transformer.

Any pair of wires with a voltage between them with a suitably low source impedance is able to source current. Your drawing doesn't really mean much to me. Incidentally, are we talking about 3-phase delta, 3-phase wye, single phase with 2 hots and a neutral, all of the above?

I can only speak for myself, but I was not talking about three phase electric power as I now realize the OP was. I guess the superposition principle still holds, but maybe the spoon thing was pushing my luck a little too far... (ahem)

I like to explain things in terms that makes it easy to visualize. 3-phase delta is a pretty easy one to answer. There are three transformer windings (secondaries) that are hooked in a series. Drawn out they appear as a triangle, hence the reason we call it delta. Each 'phase' comes off of a node from two windings. So, grab any two phases and you can see they are directly across a transformer winding. I don't see how you could not see that you can source power from any two phases. Maybe I missed the point?

foo - your red and blue picture is 'unconventional' and I think it shows that you are confused about what is happening. Arrows, such as you have drawn are usually taken to mean current flow. In that diagram, there would be no net current flow because it doesn't have anywhere to go - the two currents are in opposite directions.

Two wires with potentials which are alternating and in anti-phase will be in the same situation as if one wire is at Earth and the other is at twice that potential - the difference in potential is the same in each case.
If they are supplied with enough current to maintain them at these potentials then you can get power out of the arrangement. "Back at the supply" you could have a transformer winding which is connected to the two wires. If you connect the centre turn of this winding to Earth, the two outputs will be 180 degrees out of phase ' about Earth'.
You could also achieve this with two transformers, one for each wire and with its other output connected to Earth. The outputs could be chosen to he in phase or in anti phase- depending on which way round you connect the wires.
In a three phase supply your generator will produce three outputs, each of which is 120 degrees out of phase to the others. The generator often have three windings, connected as a 'star' or "Y" with one end of each of its windings at the centre and the other ends will have the three phases. This centre point may be connected to Earth, keeping the three phases nicely symmetrical about 0V. The 'return path' for currents flowing through a load connected across two of the phases will be via two of the windings - that was one of the original questions.

You really need to browse through Wikkers (and all the rest) to get a better idea of what's going on.